494 karma · joined January 8, 2015
89. Living out justice in the Church means purifying ecclesial relationships and structures from distortions that give rise to inequality, lack of transparency and abuse of power. In this regard, listening to the victims of spiritual, economic, institutional, sexual and power-based abuse, as well as abuses of conscience, is an integral part of a journey toward justice, which includes acknowledging the harm done, just reparation and taking steps to prevent it from happening again. Every power is at the service of communion and mission. All authority is at the service of the People of God. This ministry of service is expressed not only through our faith celebrated and lived in the Sacraments, and in the adoption of a synodal style, but also in the concrete sharing of goods. Following the example of the early Church, ecclesial resources need to be shared so that no one among us may be in need (cf. Acts 434, and so that their administration may support the mission of proclaiming the Gospel to the poorest. Regular assessments of the exercise of ministerial responsibilities should be encouraged, not as judgments on individuals, but as tools for learning and correction oriented toward mission. 116 Only to the extent that we are open to the action of the Holy Spirit will these principles of Social Doctrine become incarnate in ecclesial life. In this way, the Church will be able to bear credible witness to society that seeking the common good together, with shared responsibility and fraternity, is not a utopia, but a real possibility.
https://www.vatican.va/content/leo-xiv/en/events/event.dir.h...
https://www.congress.gov/bill/119th-congress/house-bill/7270...
"In 2010, Turchin published research using 40 combined social indicators to predict that there would be worldwide social unrest in the 2020s"
https://www.congress.gov/bill/118th-congress/senate-bill/884...
theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) :
For fun I tried it on the free model on openrouter.ai. Got the answer the first time.
https://leanprover-community.github.io/mathematics_in_lean/m...
Here's the answer just to give you a feel.
by_contra h
have h₁ : a ≠ b := h
have h₂ : |a - b| > 0 := by
apply abs_pos.mpr
exact sub_ne_zero.mpr h₁
-- Use the definition of convergence to find N₁ and N₂
have h₃ := sa (|a - b| / 2) (by linarith)
have h₄ := sb (|a - b| / 2) (by linarith)
cases' h₃ with N₁ h₃
cases' h₄ with N₂ h₄
-- Choose N to be the maximum of N₁ and N₂
let N := max N₁ N₂
have h₅ := h₃ N (by simp [N, le_max_left])
have h₆ := h₄ N (by simp [N, le_max_right])
-- Derive a contradiction using the triangle inequality
have h₇ : |s N - a| < |a - b| / 2 := by simpa using h₅
have h₈ : |s N - b| < |a - b| / 2 := by simpa using h₆
have h₉ : |a - b| < |a - b| := by
calc
|a - b| = |a - s N + (s N - b)| := by ring_nf
_ ≤ |a - s N| + |s N - b| := by
apply abs_add
_ = |s N - a| + |s N - b| := by
rw [abs_sub_comm]
_ < |a - b| / 2 + |a - b| / 2 := by
linarith
_ = |a - b| := by ring
linarithhttps://wearables.cc.gatech.edu/projects/twidor/screens.html